6 Point Estimation Example Point Estimation Suppose that we want to find the proportion p of bolts that are substandard in a large manufacturing plant To test the bolt you destroy the bolt so you do not want to check all of the bolts to see if they fail ID: 276582
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Copyright © Cengage Learning. All rights reserved.
6
Point EstimationSlide2
Example: Point EstimationSuppose that we want to find the proportion, p, of bolts that are substandard in a large manufacturing plant. To test the bolt, you destroy the bolt so you do not want to check all of the bolts to see if they fail.What is a good point estimator of p, p̂
?Slide3
Procedure: Point Estimation
Define the r.v. and determine its distribution (random sample).For the parameter of interest, determine the appropriate statistic and its formula (estimator),Calculate the statistic from the data (estimate).Suppose that bolt numbers 5, 13, 24 are substandard out of 25 bolts, what is the value of p̂?Slide4
Definition: Point EstimationA
point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic, is called the point estimator.
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Example 6.2: Point Estimation
Assume the dielectric breakdown voltage for pieces of epoxy resin is normally distributed. We want to estimate the mean μ of the breakdown voltage. We randomly check 20 breakdown voltages (below). Which point estimators could be used to estimate μ?
24.4625.6126.2526.4226.6627.15
27.3127.5427.7427.9427.9828.04
28.28
28.49
28.50
28.87
29.11
29.13
29.50
30.88Slide6
Unbiased Estimators
http://www.weibull.com/DOEWeb/unbiased_and_biased_estimators.htmSlide7
Unbiased estimator
Figure 6.1The pdf’s of a biased estimator and an unbiased
estimator for a parameter Slide8
Examples: Point EstimationFor a binomial distribution with parameters n and p with p unknown,
Is the estimator of the sample proportion , an unbiased estimator of p?For normal distribution with mean and variance 2, given a random sample of size n, X1, …., Xn
.Is the sample mean , an unbiased estimator of ?Slide9
Example 6.4: Point Estimation
Suppose that X, the reaction time to a certain stimulus, has a uniform distribution on the interval from 0 to an unknown upper limit, θ (so the density function of X is rectangular in shape, with height 1/θ for 0 x θ). It is desired to estimate θ on the basis of a random sample X1, …., Xn of reaction times.Is max(X1, …., Xn
) an unbiased estimator?XSlide10
Estimators with Minimum Variance
Figure 6.3Graphs of the
pdf’s of two different unbiased estimatorsSlide11
Principal of Minimum Variance Unbiased Estimation
Among all estimators of that are unbiased, choose the one that has minimum variance. The resulting is called the minimum variance unbiased estimator (MVUE) of .
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Estimators with Minimum VarianceIs a biased estimator always the best estimator?Slide13
Best Estimators for μ
DistrcdfBest EstimatorNormal-
< x < Cauchy- < x <
Uniform-c x – μ c
0
elseSlide14
Example 6.9( 6.2): Estimate of error
Assume the dielectric breakdown voltage for pieces of epoxy resin is normally distributed. Here s = 1.462, n = 20. What is the standard error of the best estimator of μ?Slide15
Example 6.12: Moment EstimatesLet X1, …,
Xn represent a random sample of service times of n customers at a certain facility, where the underlying distribution is assumed exponential with parameter . Estimate .XSlide16
Example: Moment EstimatesLet X1
, …., X10 represent a random sample of measurement errors of size 10 where the underlying distribution is assumed to be normal. If the observed values of the random sample areFind the moment estimates of μ and .
3.923.764.013.673.893.62
4.094.153.583.75XSlide17
Example: Moment EstimatesLet X1, ….,
X5 represent a random sample of bus wait times of size 5 where the underlying distribution is assumed to be uniform. The observed values give , , .Find the moment estimates of a and b.
XSlide18
Example 6.15: MLEA sample of ten new bike helmets manufactured by a certain company is obtained. Let X
i = 1 if the ith helmet is flawed, Xi = 0 if the ith helmet is not flawed. Assume that Xi’s are independent. p = P(a helmet is flawed) = P(Xi = 1). The observed values of x = {1,0,1,0,0,0,0,0,0,1}What is an estimate of p?
XSlide19
Example 6.16: MLELet X1
, …, Xn be a random sample from an exponential distribution with parameter . You are given the observed values x1, …, xn. Find the maximum likelihood estimate of
XSlide20
Example 6.17: MLELet X1
, …, Xn be a random sample from a normal distribution with mean μ and variance 2. You are given the observed values of x1, …, xn. Find the maximum likelihood estimate of μ and 2.
XSlide21
Example 6.18*: MLESuppose that an ecologist selects 5 nonoverlapping
regions and counts the number of plants of a certain species found in each region. Assume that all of the regions have the same area and we will set that to be ‘1’ in some units. This is a random sample of size 5 from a Poisson distribution with parameter .The observed number of plants are 2, 4, 6, 9 ,9. Find the MLE of .XSlide22
Example 6.20: Estimating Functions of ParametersIn the normal case, the MLE’s of μ
and σ2 are and .What is the MLE of the standard deviation, σ?
X